Communications in Mathematical Sciences

Volume 17 (2019)

Number 3

Energy-stable second-order linear schemes for the Allen–Cahn phase-field equation

Pages: 609 – 635

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n3.a2

Authors

Lin Wang (Beijing Computational Science Research Center, Beijing, China)

Haijun Yu (Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Beijing, China; and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China)

Abstract

Phase-field model is a powerful mathematical tool to study the dynamics of interface and morphology changes in fluid mechanics and material sciences. However, numerically solving a phase field model for a real problem is a challenging task due to the non-convexity of the bulk energy and the small interface thickness parameter in the equation. In this paper, we propose two stabilized second-order semi-implicit linear schemes for the Allen–Cahn phase-field equation based on backward differentiation formula and Crank–Nicolson method, respectively. In both schemes, the nonlinear bulk force is treated explicitly with two second-order stabilization terms, which make the schemes unconditionally energy-stable and numerically efficient. By using a known result of the spectrum estimate of the linearized Allen–Cahn operator and some regularity estimates of the exact solution, we obtain an optimal second-order convergence in time with a prefactor depending on the inverse of the characteristic interface thickness only in some lower polynomial order. Both $2$-dimensional and $3$-dimensional numerical results are presented to verify the accuracy and efficiency of proposed schemes.

Keywords

Allen–Cahn equation, energy-stable, stabilized semi-implicit scheme, second-order scheme, error estimate

2010 Mathematics Subject Classification

65M12, 65M15, 65P40

Received 14 February 2018

Accepted 17 October 2018

Published 30 August 2019